some results on the strong roman domination number of graphs

Let g=(v, e) be a finite and simple graph of order n and maximum‎ ‎degree δ(g)‎. ‎a strong roman dominating function on a‎ ‎graph  g  is a function  f‎: v (g)→{0‎, ‎1,… ,lceil ‎ δ(g)/2 rceil‎+ ‎1}  satisfying the condition that every‎ ‎vertex v for which  f(v)=0  is adjacent to at least one vertex  u ‎for which‎ f(u) ≥ 1‎+ ‎lceil frac{1}{2}| n(u) ∩ v0| rceil‎, ‎where v0={v ∊ v | f(v)=0}. the minimum of the‎ values sumv∊ v f(v), ‎taken over all strong roman dominating‎ ‎functions f of g‎, ‎is called the strong roman domination‎ ‎number  of g and is denoted by γstr(g)‎. ‎in this paper we‎ ‎continue the study of strong roman domination number in graphs‎. ‎in‎ particular‎, ‎we present some sharp bounds for γstr(g) and‎ we determine the strong roman domination number of some graphs‎.